Adaptive Petrov-Galerkin projection for stiffness reduction in multiscale thermochemical nonequilibrium kinetics

• Novel CSP-PG projection for adaptive reduced-order methods based on on-line eigen-decomposition of right-hand-side Jacobian. • Application to 1D two-temperature chemically-active non-equilibrium plasma downstream of a shock for 11 species air and 19 species N 2 − C H 4 − A r gas mixtures. • Increased Computational speed-up compared to Conventional Galerkin and Least-Square Petrov-Galerkin by up to 14 times. Stiffness decrease due to increased physics-informed integration stepping. The systems of non-linear ordinary differential equations governing reactive flows in thermochemical nonequilibrium are computationally demanding due to their high dimensionality and severe numerical stiffness, caused by the wide range of inherent kinetic spatio-temporal scales. Reduced-order models (ROMs) seek to reduce this computational cost whilst retaining sufficient accuracy. Projection-based ROMs such as Galerkin and Petrov-Galerkin (PG) have recently emerged as promising methods for compressible reactive flow simulations. However, being typically based on data-driven techniques such Proper Orthogonal Decomposition, such projection methods require offline training data, enforce a fixed reduction size and cannot guarantee a priori stability. Moreover, lacking any physical knowledge of the system dynamics, they cannot guarantee sitffness reduction. This work introduces a novel, physics-based Petrov-Galerkin ROM framework grounded in Computational Singular Perturbation (CSP) theory. The proposed CSP-PG method adaptively constructs trial and test bases online using the instantaneous eigen-decomposition of the system’s right-hand-side Jacobian. This enables dynamic tracking of the evolving time-scale structure, ensuring effective reduction in both dimensionality and stiffness through time-scale separation. Application of the CSP-PG method to a 1D two-temperature post-shock plasma flow demonstrates improved accuracy and numerical stability over standard PG projections. Although the eigen-decomposition increases the per-iteration cost, it enables adaptive time-stepping that yields an overall speedup exceeding 130%. The method demonstrates robust scalability to higher-dimensional systems, with reduction in dimensionality of up to 43%, leading to a reduction in numerical stiffness by up to two orders of magnitude.